sin^2x-tan^2x=-sin^2xtan^2x

Question

Please help

Answer 1

Use the identities:

 

sin²x + cos²x = 1

 

tanx = sinx/cosx

 

 

Simplify the left side.  The left side should match the right side.

 

sin²x - (sin²x / cos²) =

 

(sin²xcos²x - sin²) / cos²x =

 

 

sin²(cos²x - 1) / cos²x =

 

tan²x(cos²x - 1)

 

 

If        sin²x = 1 - cos²x

 

then, negating both sides of the equation by -1 to get

 

-sin²x = -(1 - cos²x)

-sin²x = cos²x - 1

 

 

Therefore,

 

tan²x(cos²x - 1) = -sin²xtan²x

 

 

Identity proves!!